Investigation Of Bifurcation Control For Nonlinear Dynamical Systems

Bifurcation control as an emerging new research field has become more and more challenging. It aims at designing a controller to modify the bifurcation properties of a given nonlinear system, and achieving some desirable dynamical behaviors.Through a complete summary and examination of the history and the actuality of the bifurcation control research, in this paper a systematic investigation into the fundamental theory and application of the bifurcation control is made by using the nonlinear vibration control theory, the nonlinear dynamics theory, the bifurcation theory. The studies have more profound theoretical significances and important engineering application values, which contribute to the development and application of the bifurcation control. The main achievements and conclusions in this dissertation are obtained as follows:1. The strategy for controlling the equilibrium bifurcation is obtained by using the open-loop control approach. The conditions of three elementary static bifurcations as saddle-node, transcritical, and pitchfork types of bifurcations for a one-dimensional ordinary differential equation are formulated. The open loop control is used to adjust the bifurcation parameter in order to obtain a desired equilibrium bifurcation diagram. By adopting the state feedback control strategy, the required bifurcations is obtained and the unwanted branches are eliminated.2. The linear and nonlinear feedback controllers are designed to control the saddle-nodes bifurcation of the forced Duffing system with the quadratic and cubic nonlinearities. In the cases of primary and superharmonic resonances, the linear feedback controller is designed to eliminate saddle-node bifurcations which would occur in the uncontrolled system, while the nonlinear one is designed to delay the occurrence of saddle-node bifurcations. Accordingly, either a linear feedback, or a nonlinear one, or a synthesis of both is adequate for the purpose of bifurcation control. Moreover, an appropriate feedback can also decrease the amplitude of the steady state response. Through the numerical simulations, the results are qualitative agreement with these of the theoretical analysis.3. The theoretical studies reveal that the designed cubic velocity feedback is affective for controlling the superharmonic resonance responses of a parametrically excited system. The amplitude of the response are reduced and the saddle-node bifurcations have are eliminated, which would take place in the resonance responses. By analyzing the bifurcation function associated with the corresponding frequency-response equation and the Jacobi matrix, the gain of the feedback control is determined. A parametrically excited oscillator with strong nonlinearity including van der Pol and Duffing type is studied for static bifurcations. The applicable range of the MLP method is extended to 1/2 subharmonic resonance systems and the bifurcation equation of a strongly nonlinear oscillator which is transformed into a small parameter system is determined by using the multiple scales method. On the basis of the singularity theory, the transition set and the bifurcation diagram in various regions of the parameter plane are analyzed. The parametrically excited pendulum with the linear and nonlinear feedback is found which validates that the controller can be designed in the engineering applications.4. The different nonlinear parametric feedback control including time is used to control the bifurcations in various nonlinear dynamical systems, such as the forced Duffing system, the Duffing-van der Pol system and the parametrically excited system. The designed controllers are testified theoretically to eliminate the saddle-node bifurcation successfully in the case of primary and superharmonic resonances and to reject the steady-state response in the case of subharmonic. The results of the numerical simulations show that the proposed feedback control method is quite effective to achieve the goal of bifurcation control.5. The coupled oscillators are studied, and the control law is obtained in the case of the one-to-one internal resonance. An approximate solution for the nonlinear differential equations is got by using the method of multiple scales. And the bifurcation analysis and the performance of the control strategy are investigated theoretically. By adjusting the control parameter, the high-amplitude periodic and chaotic motions are removed.In this paper, the innovative thinking is that the bifurcation control theory is used to investigate the nonlinear dynamical systems, which enriches the nonlinear dynamics theory and expands the nonlinear control theory. The creative things are as follows: The open-loop control approach is used to control the equilibrium bifurcation. The state feedback is extended to control...